A highly accurate high-order validated method to solve 3D Laplace - PowerPoint PPT Presentation

A highly accurate high-order validated method to solve 3D Laplace equation M.L.Shashikant, Martin Berz and Kyoko Makino Michigan State University, E Lansing, MI ,USA December 19, 2004

Approach We present an approach based on high-order quadrature and a high-order fi nite element method to fi nd a validated solution of the Laplace equation when derivatives of the solution are speci fi ed on the boundary ³ R 3 ´ ¢ = 0 in volume Ω ¡ − → ∆ ψ r ³ R 3 ´ ¢ = − ¢ on surface ∂ Ω ¡ − ¡ − → → → ∇ ψ r f r Where do we want to use this approach? In accelerator/spectrometer magnets where the magnet manufacturer pro- vides only discrete fi eld data in the volume of interest MAGNEX: A large acceptance MAGNetic spectrometer for EXcyt beams, at the Labora- tori Nazionali del Sud - Catania (Italy). (Fringe fi elds, high aspect ratio, discrete data)

What do we expect from this method/tool ? ¢ and ∂ n ¡ − ¡ − ¢ ) → → • Provide validated local expansion of the fi eld ( ψ r x i ψ r • Highly accurate (work for case with high aspect ratio) • Computationally inexpensive • Provide information about the fi eld quality and if possible reduce noise in experimentally obtained fi eld data

Note about Laplace Equation • Existence and uniqueness of the solution for 3D case can be shown using Green’s formula • Integral kernels that provides interior fi elds in terms of the boundary fi elds or source are smoothing Interior fi elds will be analytic even if the fi eld/source on the surface data fails to be di ff erentiable • Analytic closed form solution can be found for few problems with certain regular geometries where separation of variables method can be applied

Numerical methods to solve Laplace equation • Finite Di ff erence, Method of weighted residuals and Finite element methods — Numerical solution as data set in the region of interest — Relatively low approximation order — Prohibitively large number of mesh points and careful meshing re- quired • Boundary integral methods or Source based fi eld models

— Field inside of a source free volume due to a real sources outside of it can be exactly replicated by a distribution of fi ctitious sources on its surface. Error due to discretization of the source falls o ff rapidly as the fi eld point moves away from the source. ∗ Image charge method · Choose planes/grids to place point charges (or Gaussian dist) · Solve a large least square fi t problem to fi nd the charges · Lot of guess work and computation time involved in getting the solution ∗ Methods using Helmholtz’ theorem · Helmholtz’ theorem is used to fi nd electric or magnetic fi eld directly from the surface fi eld data

· In our approach we make use of the Taylor model frame work to implement this

Helmholtz’ theorem Any vector fi eld − → B that vanishes at in fi nity can be written as the sum of two terms, one of which is irrotational and the other, solenoidal − → x ) = � ∇ × � x ) + � B ( � A t ( � ∇ φ n ( � x ) x s ) · − → ∇ · − → Z Z � x ) = 1 � n ( � B ( � x s ) ds − 1 B ( � x v ) φ n ( � x v | dV 4 π | � x − � x s | 4 π | � x − � ∂ Ω Ω x s ) × − → ∇ × − → Z Z � x ) = − 1 � n ( � B ( � x s ) ds + 1 B ( � x v ) � A t ( � dV 4 π | � x − � x s | 4 π | � x − � x v | ∂ Ω Ω

∇ × − → ∇ · − → For a source free volume we have, � x v ) = 0 and � B ( � B ( � x v ) = 0 x ) and � Volume integral terms vanish, φ n ( � A t ( � x ) are completely determined from the normal and the tangential magnetic fi eld data on surface ∂ Ω x s ) · − → x s ) × − → R R � n ( � B ( � x s ) � n ( � B ( � x s ) x ) = 1 x ) = − 1 � φ n ( � ds A t ( � ds ∂ Ω ∂ Ω 4 π 4 π | � x − � x s | | � x − � x s | � B is Electric or Magnetic fi eld ∂ Ω is a surface which bounds volume Ω � x s and � x v denote points on ∂ Ω and within Ω � ∇ denote the gradient with respect to � x v � n is a unit normal vector pointing away from ∂ Ω

Implementation using Taylor Models • Split domain of integration ∂ Ω in to smaller regions Γ i • Expand them to higher orders in surface variables � r s and the volume variables � r — Expanded in � r s about the center of each surface element — Expanded in � r about the center of each volume element — Field is chosen to be constant over each surface element

Z x Nx Z y Ny i x = N x − 1 ,i y = N y − 1 ,k x = ∞ ,k y = ∞ X g ( x, y ) dxdy = x 0 y 0 i x =0 ,i y =0 ,k x =0 ,k y =0 h 2 k y +1 h 2 k x +1 y x (2 k y + 1)! · 2 2 k y (2 k x + 1)! · 2 2 k x à ! y i y +1 + y i y x i x +1 + x i x g 2 k x , 2 k y , 2 2 We can obtain: � f ( � r s ) Scalar potential φ n ( � r ) if we choose g ( x, y ) = � n s · | � r − � r s | � f ( � r s ) Vector potential � A t ( � r ) if we choose g ( x, y ) = � n s × | � r − � r s |

• — Bene fi ts ∗ The dependence on the surface variables are integrated over sur- face sub-cells Γ i , which results in a highly accurate integration formula ∗ The dependence on the volume variables are retained, which leads to a high order fi nite element method ∗ By using su ffi ciently high order, high accuracy can be achieved with a small number of surface elements • Depending on the accuracy of the computation needed, we choose step sizes, order of expansion in r ( x, y, z ) and order of expansion in r s ( x s , y s , z s )

Validated Integration in COSY x iu Z ³ ³ ´ ³ ´ ´ f ( � x ) dx i ∈ P n,∂ − 1 f � x | x i = x iu − x i 0 − P n,∂ − 1 f � x | x i = x il − x i 0 , I n,∂ − 1 f x il This method has following advantages: * No need to derive quadrature formulas with weights, support points x i , and an explicit error formula * High order can be employed directly by just increasing the order of the Taylor model limited only by the computational resources * Rather large dimensions are amenable by just increasing the dimensionality of the Taylor models, limited only by computational resources

An Analytical Example: the Bar Magnet x 1 ≤ x ≤ x 2 , | y | ≥ y 0 , z 1 ≤ z ≤ z 2 As a reference problem we consider the magnetic fi eld of rectangular iron bars with inner surfaces ( y = ± y 0 ) parallel to the mid-plane ( y = 0)

From this bar magnet one can obtain analytic solution for the magnetic fi eld � B ( x, y, z ) of the form ⎡ ⎛ ⎞ ⎛ ⎞ ⎤ X ⎝ X i · Z j ⎝ X i · Z j B y ( x, y, z ) = B 0 ( − 1) i + j ⎠ + arctan ⎣ arctan ⎠ ⎦ Y + · R + Y − · R − 4 π i,j ij ij ⎡ ⎛ ⎞ ⎤ ⎝ Z j + R − X B x ( x, y, z ) = B 0 ij ( − 1) i + j ⎣ ln ⎠ ⎦ Z j + R + 4 π i,j ij ⎡ ⎛ ⎞ ⎤ ⎝ X j + R − X B z ( x, y, z ) = B 0 ij ( − 1) i + j ⎣ ln ⎠ ⎦ X j + R + 4 π i,j ij

where i, j = 1 , 2 , X i = x − x i , Y ± = y 0 ± y, Z i = z − z i ³ ´ 1 X 2 i + Y 2 j + Z 2 2 and R ± = ± We note that only even order terms exist in the Taylor expansion of this fi eld about the origin.

BY 12 10 8 6 4 2 0 -0.003 -0.002 -0.001 0 0.004 X 0.001 0.002 0 0.002 -0.002 0.003 Y -0.004 B y component of the fi eld on the mid-plane.

Results 1. To study the dependency of the Interval part of the potentials and � B fi eld on the surface element length • All of the volume is considered as just one volume element • Examine contributions of each surface element towards the total integral — Expansion is done at � r = ( . 1 , . 1 , . 1) and — Plot of interval width VS surface element length for scalar po- tential — Plot of interval width VS surface element length for vector po- tential ( x component)

• Plot of interval width VS Order for di ff erent surface element length for x component of Magnetic fi eld

Interval Width VS Surface Element Length for Scalar Potential -2 Order 8 Order 7 -4 Order 6 Order 5 Order 4 Order 3 -6 Order 2 -8 -10 LOG10(Interval Width) -12 -14 -16 -18 -20 -22 -13 -12 -11 -10 -9 -8 -7 -6 -5 -4 -3 LOG2(Surface Element Length) Figure 1: Integration over single surface element (for φ )

Interval Width VS Surface Element Length for X Vector Potential -2 Order 8 Order 7 -4 Order 6 Order 5 Order 4 Order 3 -6 Order 2 -8 -10 LOG10(Interval Width) -12 -14 -16 -18 -20 -22 -13 -12 -11 -10 -9 -8 -7 -6 -5 -4 -3 LOG2(Surface Element Length) Figure 2: Integration over single surface element (for A x )

Interval Width VS Order for different stepsize -4 Step size = 0.0166 Step size = 0.0192 Step size = 0.0147 -5 -6 -7 LOG(Interval Width) -8 -9 -10 -11 4 4.5 5 5.5 6 6.5 7 7.5 8 ORDER Figure 3: Interval width VS Order (for di ff erent step size)

2 Study the dependency of the Polynomial part and Interval part of the B fi eld on the volume element length • The surface element length is locked at 1/128 • Plot of the error calculated for the polynomial part VS the volume element length • Plot of interval width VS volume element length for y component of Magnetic fi eld

Interval Width VS Length of Volume Element for Scalar Potential -1 Order 8 Order 7 -2 Order 6 -3 -4 -5 LOG(Interval Width) -6 -7 -8 -9 -10 -11 -7.5 -7 -6.5 -6 -5.5 -5 -4.5 -4 -3.5 -3 LOG2(Length of Volume Element) Figure 4: Interval width VS Volume element length (for φ )